A model for determining initial dislocation-densities associated with martensitic transformations

Introduction

Lath martensitic steels due to their high strength and toughness have myriad military and civilian applications. These properties are uniquely inherent to martensitic steels due to its lath microstructure that has distinct orientations, distributions, and morphologies pertaining to martensitic transformations.1

In previous investigations, we have developed a dislocation-density crystalline framework to investigate how deformation and failure evolves in martensitic steels.2 The formulation is based on accounting for variant morphologies and orientation, retained austenite, and initial dislocations densities that are uniquely inherent to martensitic microstructures. One of the challenges in accurately modelling lath martensite is determining the initial dislocation-densities associated with the martensitic transformation, since it can affect the overall behaviour of martensitic structures.1 Sandvik and Wayman 3 4 3−5 classified dislocations in martensitic microstructure as being of two basic types, transformation and interface dislocations. The transformation dislocations are screw dislocation in all four 〈111〉 _{α ′} directions, with a/2 [ 1] _{α ′} as the dominant direction. 3 4 3−5 It is also difficult to obtain the initial dislocation-densities from experimental data,6 since the specific slip-systems, and how these dislocation-densities are apportioned between slip-systems are difficult to ascertain.

In this paper, we used a dislocation-density based crystalline plasticity formulation and specialised finite element techniques to predict the initial mobile and immobile dislocation-densities associated with martensitic transformations. The major motivation for this work is based on accounting for the initial dislocation-densities that inherent to martensitic steels. The proposed approach is based upon modelling the shear part of a martensitic transformation and then incorporating it within a multiple slip dislocation-density based crystal plasticity approach.

Dislocation-density based multiple-slip constitutive formulation

The formulation for the multiple slip crystal plasticity rate dependent constitutive relations, and the derivation of the evolutionary equations for the mobile and immobile dislocation-densities, which are coupled to the multiple-slip crystalline formulation,7 are outlined here.

It is assumed that the velocity gradient can be decomposed into a symmetric part, the deformation rate tensor, D_ij and an anti-symmetric part, the spin tensor, W_ij. It is further assumed that the total deformation rate tensor, and the total spin tensor, D_ij, can be then additively decomposed into elastic and plastic components as (1a) (1b) where W_ij includes the rigid body spin. The inelastic parts are defined in terms of the crystallographic slip rates as (2a) (2b) where α is summed over all slip-systems, and and are second order tensors, and are defined in terms of the unit normals to the slip planes and the unit slip vectors to the slip directions.

For rate dependent inelastic materials, the constitutive description on each slip-system can be characterised by a power law relation as (3) where is the reference shear strain rate which corresponds to a reference shear stress, and m is the rate sensitivity parameter. The reference stress relate to a square root of immobile dislocation-density as (4) where G is the shear modulus, b is the magnitude of the Burgers vector, is the static yield stress, and the coefficients, a_ξ are interaction coefficients, and generally have a magnitude of unity.

Phenomenological transformation model

The austenite phase is represented as an fcc. structure with twelve potential slip-systems with habit planes of {111} and slip directions of 〈110〉.9 The martensitic phase is represented as (bct/bcc) with 24 potential slip-systems with {110} and {112} slip-planes, and slip directions of 〈111〉.10

In this study, we utilise the single shear transformation theory introduced by Otsuka and Wayman11, Bhadeshia,12 and Bowles and Mackenzie.13 This approach is based on having a glissile interface between parent and product phases. This can exist if one direction and one plane (habit plane) is unrotated and undistorted. Hence, this martensitic transformation (P₁) is an invariant plane strain (IPS), and can be described by three successive transformations (see Fig. 1), not necessarily ordered as:

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Figure 1

Martensitic transformation as described by the three transformations: B, bain strain; R, a rigid body rotation; P₂, simple shear

Both (P₁), and (P₂) are IPS, where each can be represented as P = I+ d′×p, where p is normal to the invariant plain, and d is the deformation vector. The product of both IPSs is an invariant line strain, and is given by (7) Based on calculations by Otsuka and Wayman,11 and Bhadeshia12 for (S, P₁, P₂) for the Kurdjumov−Sachs (K−S) 24 variants, we obtained the following values for the deformation tensors and the normals to the simple shear plane (d₂ and p₂) relative to the parent and transformed structures coordinates as (8)

Computational techniques

The total deformation rate tensor, D_ij, and the plastic deformation rate tensor, are needed to update the material stress state. The method used here is the one developed by Zikry14 for rate dependent crystalline plasticity formulations.

Results and discussion

The multiple slip dislocation-density based crystal plasticity formulation and the specialised finite element algorithm were used to investigate the martensitic transformation of steel alloys. The martensite orientation and transformation, and the values of (d₂) and (p₂) are represented as outlined above. An orientation matrix, T, for the transformed matrix, was calculated as (9) where is the normalised value of d₂.

The material properties (Table 1) that are used here are representative of low nickel alloy steel.

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Table 1

Properties of martensite and austenite steels

Properties	Martensite phase	Austenite phase
Young's modulus E	228 GPa	200 GPa
Static yield stress σy	517 MPa	367 MPa
Poisson's ratio ν	0·3	0·3
Rate sensitivity parameters m	0·01	0·01
Reference strain rate	0·001 s−1	0·001 s−1
Critical strain rate	104 s−1	104 s−1
Burger vector b	3·0×10−10 m	3·0×10−10 m
Initial immobile dislocation-density	1010 m−2	1010 m−2
Initial mobile dislocation-density	107 m−2	107 m−2

Using the method outlined by Kameda and Zikry,15 the initial coefficient values, needed for the evolution of the immobile and mobile densities, for equations (5) and (6), were obtained as (10) We modelled a simple shear problem for an invariant plane strain for both an fcc and a bcc single crystal. For the fcc model, four slip-systems are active: , , and . All have the identical mobile dislocation-density of 0·14×10¹⁴ m⁻², and immobile dislocation-density of 0·64×10¹⁴ m⁻² (Fig. 2). The sum of the dislocation-densities is close to the values extrapolated from experiments6 of 3·1×10¹⁴ m⁻². However, equal values of dislocation-densities on different active slip-systems are not observed experimentally, and a dominant slip-system is usually observed. 3 4 5 3−6

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Figure 2

Immobile and mobile dislocation-densities for most active slip system during martensitic transformation ×10¹⁴; for slip systems , , and

Hence, we then modelled the same problem for a bcc structure. For the bcc model, even though we initially assumed that all 24 slip-systems are potentially active, after deformation for the shear problem with the IPS conditions, only one slip system is active with a mobile dislocation-density of 0·46×10¹⁴ m⁻² and an immobile dislocation-density of 2·57×10¹⁴ m⁻² (Fig. 3). This active slip-system is parallel to the long direction of laths, and match the most active slip-system reported by Otsuka and Wayman.11 The sum of the immobile and mobile dislocation-densities on bcc crystal is almost the same as that experimentally obtained.6

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Figure 3

Immobile and mobile dislocation-densities for most active slip system during martensitic transformation ×10¹⁴; for slip system

Hence, the transformation dislocation-densities can be interpreted as a combination of dislocation-densities evolving in both fcc and bcc crystals. The ratio of the contribution of each slip system can be correlated with experimental observations. However, for simplicity, we can use a fully bcc system to determine the initial immobile and mobile dislocation-densities for modelling lath martensite in steels. This is also physically consistent with the observed dominance of one active slip system in martensitic transformation in lath steel. 3 4 5 3−6

Summary

In summary, a multiple-slip dislocation density based crystal plasticity model and specialised finite element techniques were used to predict initial immobile and mobile dislocation-densities for an IPS shear deformation related to martensitic deformations. The predicted values for the fcc and bcc structures are almost identical as those obtained by Morito et al.6 While four equally activated slip-systems are obtained for transformation in fcc crystals, only one slip-system is dominant for the bcc crystals, which is consistent with the experimental observations. 3 4 5 3−6 These initial values, which are based on slip-system activity and orientations, can be an important step in accurately modelling martensitic structures on the microstructural scale.